convert a .fbx model to threejs with textures
Okay, so thanks to Duarte for the direction. Here's what I ended up doing for anyone having a similar issue.

I started by importing the FBX into blender, then joined all the objects so I had a single large object.

Then I added a decimate modifier, made a copy of the object and removed all materials from the copy, after which I added a new blank material.

To end up with a single texture I made a new blank image in the UV Editor (size 4096x4096) and made a smart UV projection of the texture-less model. Then I baked the textures from the textured model into a single image and assigned it to the blank image used as a texture on the second model.

This video helped a lot.

I'm still working on the export, but I'll post an edit when I have the process pinned down.

EDIT: I ended up exporting to collada. It has an efficient loader in threejs and is easy to work with. Blender has a built in exporter.